Copyright Alan Whitehead & Earthschooling: No Part of this book, post, URL, or book excerpt may be shared with anyone who has not paid for these materials.
Alan speaks in a very symbolic and esoteric manner in some parts of his books. Although they can be read anthroposophically, passages speaking of Atlantis, archangels, gods, etc. do not need to be taken literarily to be meaningful. The more you read, the more you will realize he uses many different religions to express ideas in a symbolic manner and not in a religious manner. His writings are not religious. In some places his writings are meant to refer to religious events in a historical way. In some places he is using religious figures (from Christianity, Judaism, Islam, Buddhism, Hinduism, Paganism, Ancient Roman and Greek Religions, etc.) in a symbolic manner. However, at no point is he promoting a specific religion or speaking from a religious point of view.
I have kept the writing as close to one-hundred percent original so you will also find that he speaks of Australia often and some spelling or manners of speaking may be cultural. Any words I have changed are presented like this: <word>.
Also keep in mind that these books are written by a Waldorf teacher with decades of experience who also studied with a Steiner student himself, so he speaks to an audience that is dedicating their lives to the Waldorf method without exception.
Because of this, all of his views are not reflected in the Earthschooling curriculum and not all of them may be ones you want to embrace or are able to use. In all of Alan Whitehead’s writings the opinions are his own and may not align with Earthschooling or Waldorf Books. In some cases, we will be updating some of these chapters in the future with additional and/or updated information.
Ultimately, however, as I read through these passages I find I can distill wisdom from even those paragraphs that do not resonate with me.
We invite you to read with an open mind and heart and with eagerness to learn and discuss…
MATHS IS JUST A GAME
Borrowing & Carry-over – Class 3 – Main Lesson
“Darling, call the children for tea will you?” yelled Mother Sensible through the steam of the kitchen. It was a lot to ask her husband, resting quietly on the lounge; after all, he had been working all day! “They’re out on the verandah – they’ve been there for an hour – what can they be up to?” he replied helpfully.
The twins were doing their homework; inventing a number game. In this case it was on a dart board. However they had abandoned the time-honored rules of darts, and created a whole new set, based on one sibling calling out a mathematical instruction just as the dart was thrown by the other.
“…and on whatever number the dart lands, I have to do the process.” Said Tan later at the table “For instance, I was throwing, and Lily yelled ‘Times 3!’. My dart landed on 20, so I had to work out – quickly – 20 x 3, which is…er,” Tan looked down at his fingers. Dad looked down in sympathy.
“60!” yelled Lily “And I didn’t’ need to count on my fingers!”
“Nyyl said it was okay to use fingers – and toes even – for number work, when you’re young anyway.”
“I wasn’t allowed to at school.” Said Mother as she dished out the vegetables “I used to through; I got quite quick at it really.”
“They say that it aids muscular, or unconscious, memory to engage the body in calculation. Dad surveyed his family’s reaction to this bit of numerical wisdom – after all, he was a surveyor!
“I like the sound of your dart board game,” he continued “but don’t forget to bring the board home when you’re finished with it; I like the odd game myself you know. I wonder if it’s because darts – and other ‘number’ games, like bobs and billiards – keep one’s calculation faculties honed? Hmmm, so what is this new maths main lesson? It sounds as if you’re all just playing games to me.”
“That’s it!” the twins chorused “Number Games.” Lily went on “it’s fun; I was kind of scared of a 3-week maths main lesson, but gee, we’ve played – er, done, so many games, all with numbers in them. For instance, we were each given one of those 15-piece number puzzles – the ones where you move the little tiles.”
“I like those; there’s always one gap. You can make all sorts of sums with them straight away – no writing.” Said Tan returning from the kitchen with the tomato sauce “Nyyl says you can make two hundred billion number combinations with that simply ‘toy’ – or is that two thousand million?”
“Same think; so you’re learning about big numbers are you? Class 3 sounds about the right time.” Mother had some tomato sauce too; mind you, she still thought that it wasn’t necessary, the meal being tasty enough. Lily caught the discussion ball.
“Big numbers yes, we play a game with big numeral cards. Each child takes a card and we all line up to make the big numbers. It’s funny, today we made 47,624,000, or something like that. I was the 6, or 600,000 – in the 100,000 column; but when I stood out, instead of the number being 600,000 short, it was nearly 40,000,000 less! Wasn’t it Tan?”
“Yeah, numbers sure are funny things; I liked it when we had the problem and answer cards, and split into 2 teams. One group would make up a sum on their cards – the others would make up the answer. It’s weird having to make up sums, usually the teacher does it – it’s harder than you think, to make it work that is.”
“I daresay,” responded Dad abstractly “but all this reminds me of some of the number games we used to muck around with in college – the Visible Number Series. There was the, um, the Triangle numbers, and the Square…”
“…and the Rectangle numbers…”
“…and the Gnomes!”
“Gnomes?” Mother and Dad turned to their son in puzzlement.
“Yes, Gnomes – they’re all odd numbers, and they arrange in the form of a right angle, like a carpenter’s square.”
“Oh Gnomons!” exclaimed Dad in relief “I forgot about those. They have most interesting properties. And while on odd and even; working out of the basis of the 4 operations, there are some pretty ‘odd’ number patterns here. An odd multiplied by an odd always give an odd – even x evens give evens. But the inverse function, divide, gives odd divided by odd still always gives odd, but even divided by even can give odd oreven.
Mother had been quickly working some out herself – on her fingers! “Even added to even always give even; yet odd added to odd always, curiously enough, always gives even as well! Even minus even is always even; and odd minus odd is always even!”
“We draw gnomes…gnomons, in our books (I put a little red, pointed hat on one of mine – hee, hee).” Giggled Lily as she reached for the tomato sauce “you could see the number patterns really easily. We also did a Prime Number pattern – and the multiplication table; some of the kids found even more interesting number sequences than we did last year in our Times Tables main lesson. It’s really easy to find the ‘square’ numbers with the 12 across – 12 down chart; say 4 down an d4 across gives 16, a square number. There’s a lot of odd/even stuff in these as well.
But my favorite is the Magic Squares, those numbers are just amazing. One of them added up to 9 on everyrow across, and every row down, and, what was really amazing, on both diagonal rows! Nyyl had some books on magic squares – I love them.”
“That was neat,” joined Tan eagerly “but what about the other number patterns? Get this one – 1 x 8 + 1 = 9 right? Then 12 x 8 + 2 = 98, get it? No? 123 x 8 + 3 = 987 – aw, come on. I’ll only do one more 1234 x 8 + 4 = 9876 – it goes on like this forever!
You might do better with one of the 9 patterns, try this one – 9 + 9 = 18, 1 + 8 = 9. 9 + 9 + 9 = 27, 2 + 7 = 9. You see, the sum of the sum always equals 9, no matter how big the number. Let’s try a big one, we did this at school, that’s why I can remember. One hundred and forty-six 9s are 1314, 1 + 3 + 1 + 4 is 9. Who designed it all – God?!”
Dad looked up at the simplicity yet sublimity of this innocent utterance, which led his mind back to his own school days, when thoughts like that drifted around the fringes of his consciousness.
“I remember a weird number pattern, it went something like 3 x 37 = 111, 6 x 37 = 222, 9 x 37 = 333 – ad infinitum.”
“What?” said a blond and a black head together.
“You know, forever – infinity. Infinity is as hard a concept to grasp as naught. Oddly enough, the symbol of infinity is the same as that of the astral body, the curve of multiplication, the Lemniscate itself – and the symbol of naught is the curve of addition, that of the physical body – the Circle. Never mind, just idle speculations – did you learn the Pick a Number games?”
“You bet, pick a number.” Said Tan, his eyes narrowing.
“Er, 20 – sorry, I’m not supposed to tell you am I? Okay, I’ve got one (12).”
“Now times it by 3 (36), divide by 2 (18), times 3 again (54), divide by 2 again (27) – now divide by 9 – what’s the answer?”
“3.” Said Dad, his smile secretive.
“Then your original number was 12 – wasn’t it!?”
“No? Let me see, it was 102…okay, only joking, 12 it was. But how did you know?”
“With this Pick a Number, you just have to times the answer, 3, by 4 to get the original, 12 – smart eh? We did lots of those.”
“Alright, now it’s my turn.” said Mother staring into astral, er, infinite, space “If I had a chessboard of 64 squares, and I placed 2 grains of wheat on the first square, multiply by 2 for the second square – that’s right, 2 x 2 = 4. You’ve been looking at square numbers, then If you square 4, that’s 16 on the next ‘square’ of the board – and so on right to the 64th square. How many grains of wheat would there be on the board?”
“That’s too hard, but I’d say um, a hundred?” – “Maybe thousands?” said the twins unsurely.
“Actually there are millions and millions; enough wheat to feed the whole world for a year or more. That’s the power of square or ‘power’ numbers”. Mother sat back to enjoy her triumph.
The main lesson being described by the twins contained yet another important learning area: Borrowing, or ‘paying back’.
Nyyl had proceeded with seemly caution here, because the difficulty of understanding this arcane and abstract process in the 4-operation algorithms was very real.
In the early stages, numbers were taught referring always to things; I borrow 10 Apples – and pay 10 apples back in a subtraction sum. Money calculation fitted in well here; money is very real, even – or especially – to 9-year-old children. The image of moving cents and dollars around the columns, rather than mere numbers, added an element of reality.
Some explanatory work calculating with the linear measure metric numbers from a previous main lesson was done. Here we have 4 numerical factors, all with different borrowing, pay-back and carrying facilities – 10 millimeters in a centimeter, which has 100 in a meter, which in turn has 1000 in a kilometer. This variety of ‘things’ to calculate enhances mobility of thinking and breadth of understanding. Care was always taken that in the exercises on the board, there were at least 3 levels of difficulty (or ease, depending on the child’s skill level). Combined with this, Nyyl also applied that age-old technique of restraint; holding the horses on a tight rein, in the early stages at least.
The child’s request to move up to the next ‘level of difficulty’ would be refused, at least twice. This is to protect them from their own unrealistic expectations – or indeed ambitions. Secondly, it is more important to build up a ‘head of steam’ in the Will. The child’s will should not be stretched or over-extended; rather stimulated by the frustration (mild only of course) of restraint.
When the secretly smiling teacher at last reluctantly accedes to the clamoring demand to do ‘harder sums!’, the will surges into the task with genuine enthusiasm. Children – or anyone for that matter – would rather push than be pushed.
This principle extends even to classroom explanation; whilst holding forth on the mysteries of the more advanced carry-over processes in, say, a division sum, the teacher might insist that the children in the ‘easier’ groups do not listen. This way one can be sure most of them will!
“…and then we did clapping number games.” Tan was by now unstoppable “we would have to answer mental arithmetic questions by clapping the answers – sometimes with the class, at others, alone. It is was a big number, we would clap the tens over our heads; and the 100s behind our backs – give me a sum.”
“Ahh…11 times 11.” Said a bemused father. Tan clapped once behind his back, twice above this head, and one normal clap.
“121 yea! Nyyl said that this was a kind of living abacus, you know, those wooden frames you see Chinese shopkeepers us. Nyyl said we shouldn’t use them, they lead to ‘mechanical calculation’ rather than living – whatever that means.”
“They might be ‘mechanical’, but they sure are fast. I wonder what Nyyl thinks about using computers for calculating. These make the abacus appear positively organic! I understand that you don’t use electronic aids for sums and whatnot until high school – a good policy I’d say, that way the various processes become part of your being. But I like that whole-body activity; and you’ve been learning about the number columns have you?”
“You bet.” – “Yes,” said Tan and l=Lily, the latter continued “Nyyl said that they were like the columns of a temple; the temple we drew had 10 columns – see, like this.” She held up her hands, ten fingers outstretched “Okay Mum, what’s the 1st column?”
“The units of course; do you think I’m an innumerate or something?!” she smiled, but this faded with the next question.
“Okay then, what’s the 10th finger, er, number column?”
“Er, hundred thousan…millions!” Mother began desperately counting on her fingers – under the table of course!
“The 10th finder’s the billions column!” shouted Tan.
“You nuisance – I was almost there.” Mother’s expression went whimsical as she began to hum “There is a temple…out of…”
“What’s the sone Dear?” said Dad with interest.
“Oh just a ditty we learnt at school – our teacher wrote it as I recall. It helped us remember the number columns…held up with columns straight and tall…dum de dum.”
“Sung it Mum.” – ‘Yes, sing it.” They chimed.
“Go on Dear, sing it. As we don’t have a T.V. (thank the Archai!), we have to make our own entertainment – even if it’s only a maths jingle.” Mother glared at her husband’s gratuity, but then smiled shyly, breathed deeply, and began to sing in the sweetest of tones. All eating was, believe it or not, suspended.
There is a temple with its roof out of sight
Held up by columns straight and tall – to the right
Of them all are the units so small
Left of them is the columns of ten, lots of tens
Column number 3, from the right side you see
Are the 100s – the 1000s are 4th, they are 4th
Ten thousands are found in the next column long
By the 100,000s, so mighty and strong
Beside the millions they belong – and there are more
But they’re too long – so I think I’ll end this song!
(Full music notation in my book 33 Sun Songs)
The small rendition was finished, she blushed at the beaming faces of her appreciative family, who burst into genuine applause.
“I think their response says it all;” said Dad, Lily looked up.
“Treasure? We’re having a maths treasure hunt tomorrow. The clues are in little boxes, but on the lids there are borrowing sums to do – some easy, some very hard. If you get the sum right, you can look inside to find the next clue; we ‘hunt’ in teams. That’s right isn’t it Tan?”
“What? Ugh…I think so; yest I’m sure of it now.” The family looked to him for elaboration, though he couldn’t imagine why.
“The chips do need tomato sauce.”
Leave a Reply